On the residual dependence index of elliptical distributions

TL;DR

This paper investigates the residual dependence indices of multivariate elliptical distributions, especially under Gumbel and Weibull tail assumptions, to better understand tail dependence in extreme value modeling.

Contribution

It introduces formulas for partial residual dependence indices of elliptical distributions with specific tail behaviors, extending tail dependence analysis beyond Gaussian cases.

Findings

Derived formulas for residual dependence indices under Gumbel max-domain of attraction.

Discussed estimation methods for Weibull-tail distributions.

Enhanced understanding of tail dependence in elliptical models.

Abstract

The residual dependence index of bivariate Gaussian distributions is determined by the correlation coefficient. This tail index is of certain statistical importance when extremes and related rare events of bivariate samples with asymptotic independent components are being modeled. In this paper we calculate the partial residual dependence indices of a multivariate elliptical random vector assuming that the associated random radius is in the Gumbel max-domain of attraction. Furthermore, we discuss the estimation of these indices when the associated random radius possesses a Weibull-tail distribution.